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correctness_of_formula_propagation [2007/04/14 17:09] vkuncak |
correctness_of_formula_propagation [2007/04/14 17:19] vkuncak |
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| Using an example of a single loop, we explain that if the analysis terminates then it computes an inductive invariant weaker than the initial formula. | Using an example of a single loop, we explain that if the analysis terminates then it computes an inductive invariant weaker than the initial formula. | ||
| + | |||
| + | By induction, to prove that property holds after any number of loop iterations, we show that it holds initially and that it is preserved by each loop iteration. | ||
| Consider a loop | Consider a loop | ||
| Line 21: | Line 23: | ||
| \mbox{fact}_i(w) | \mbox{fact}_i(w) | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Let $I$ denote $\mbox{fact}_i(w)$. We have shown that $I$ is preserved across every loop iteration. Note that $sp(I,\mbox{assume}(F);c)) \rightarrow I$ corresponds to | + | Let $I$ denote $\mbox{fact}_i(w)$. We have shown |
| + | \begin{equation*} | ||
| + | sp(I, \mbox{assume}(F);c)) \rightarrow I | ||
| + | \end{equation*} | ||
| + | That is, $I$ is preserved across every loop iteration. Note that $sp(I,\mbox{assume}(F);c)) \rightarrow I$ corresponds to | ||
| \begin{equation*} | \begin{equation*} | ||
| I \rightarrow wp(I, \mbox{assume}(F);c) | I \rightarrow wp(I, \mbox{assume}(F);c) | ||